A Characterization of Planar Mixed Automorphic Forms

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Volume 2011, Article ID 239807,9pages doi:10.1155/2011/239807

Research Article

A Characterization of Planar Mixed

Automorphic Forms

Allal Ghanmi

Department of Mathematics, Faculty of Sciences, Mohammed V University, P.O. Box 1014, Agdal, Rabat 10000, Morocco

Correspondence should be addressed to Allal Ghanmi,[email protected]

Received 29 December 2010; Accepted 5 April 2011

Academic Editor: Charles E. Chidume

Copyrightq2011 Allal Ghanmi. This is an open access article distributed under the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We characterize the functional space of the planar mixed automorphic forms with respect to an

equivariant pair and given lattice as the image of the Landau automorphic formsinvolving special

multiplierby an appropriate isomorphic transform.

1. Introduction

Mixed automorphic forms of typeν, μarise naturally as holomorphic forms on elliptic vari-eties 1 and appear essentially in the context of number theory and algebraic geometry. Roughly speaking, they are a class of functions defined on a givenHermitian symmetric spaceXand satisfying a functional equation of type

Fγ·xjνγ, xjμργ, τxFx, 1.1

for everyx ∈ X andg ∈ Γ. Here,jα_{γ, x}_;_α_∈ _{is an automorphic factor associated to an}

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In this paper, we are interested in the space of mixed automorphic forms Mν,μ_Γ,τ

defined on the complex planeX with respect to a given latticeΓin and an equivariant

pairρ, τ. We find thatMν,μ_Γ,τis isomorphic to the space of Landau automorphic forms6,

Fzγχτ

γjBν,μτ _{γ, z}_F_z_; _z∈

, γ ∈Γ, 1.2

of “weight”

Bτν,μz νμ

∂τ_∂z

2

−∂τ

∂z

2

Bτν,μ, 1.3

with respect to a special pseudocharacterχτ defined onΓand given explicitly through5.3

below. The crucial point in the proof is to observe that the quantityBτν,μis in fact a real constant

independent of the complex variablez.

The exact statement of our main resultTheorem 5.1is given and proved inSection 5. In Sections2 and 3, we establish some useful facts that we need to introduce the space of planar mixed automorphic formsMν,μ_Γ,τ. We have to give necessary and suﬃcient condition

to ensure the nontriviality of such functional space. InSection 4, we introduce properly the functionϕν,μτ that serves to define the pseudocharacterχτ.

2. Group Action

Let

GT

g: a, b≡

a b

0 1

; a∈T, b∈

2.1

be the semidirect product group of the unitary groupT {λ ∈;|λ| 1}and the additive

group,. Gacts on the complex plane by the holomorphic mappingsg·z azb;

g a, b,z∈, so that can be realized as Hermitian symmetric space G/T.

By aG-equivariant pairρ, τ, we mean thatρis aG-endomorphism andτ : → is

a compatible mapping, that is,

τg·zρg·τz; g∈G, z∈. 2.2

Now, for given real numbers ν, μ, and an equivariant pairρ, τ, we define Jρ,τν,μ to be the

complex-valued mapping

Jρ,τν,μ

g, z:jνg, zjμρg, τz; g, z∈G×, 2.3

wherejα_;_α_∈ _{is the “automorphic factor” given by}

jαg, ze2iαz,g−

1_·0

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Here and elsewhere,zdenotes the imaginary part of the complex numberzandz, wzw

the usual Hermitian scalar product on. Thus, one can check the following.

Proposition 2.1. The mappingJρ,τν,μsatisfies the chain rule

Jρ,τν,μ

gg, ze2iφν,μρ g,g_Jν,μ ρ,τ

g, g·zJρ,τν,μ

g, z, 2.5

whereφν,μρ g, gis the real-valued function defined onG×Gby

φρν,μ

g, g: ν

g−1·0, g·0μρ g−1·0, ρg·0. 2.6

Proof. For everyg, g∈Gandz∈, we have

Jρ,τν,μ

gg, zjνgg, zjμρgg, τz

jνgg, zjμρgρg, τz.

2.7

Next, one can see that the automorphic factorjα_·_,_·_satisfies

jαhh, we2iαh−

1_·0,h_·0

jαh, h·wjαh, w, 2.8

for everyh, h∈Gandw∈. This gives rise to

Jρ,τν,μ

gg, ze2iφρν,μg,g_jν_{g, g}·_z_jμ_ρ_g_{, ρ}_g·_τ_z_Jν,μ ρ,τ

g, z. 2.9

Finally,2.5follows by making use of the equivariant conditionρg·τz τg·z.

Remark 2.2. According toProposition 2.1above, the unitary transformations

Tν,μ

g f

z:Jρ,τν,μ

g, zfg·z 2.10

for varyingg ∈Gdefine then a projective representation of the groupGon the space ofC∞ functions on.

3. The Space of Planar Mixed Automorphic Forms

LetΓbe a uniform lattice of the additive group,that can be seen as a discrete subgroup

ofGby the identification

γ∈Γ −→1, γ

1 γ

0 1

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so that the action ofΓon is the one induced from this ofG, that is,

γ·zzγ. 3.2

Associated to suchΓand given fixed data ofν, μ >0 andρ, τas above, we performMν,μ_Γ,τ

the vector space of smooth complex-valued functions F on satisfying the functional

equation

Fγ·zJρ,τν,μ

γ, zFz jνγ, zjμργ, τzFz, 3.3

for everyγ ∈Γandz∈.

Definition 3.1. The spaceMν,μ_Γ,τis called the space of planar mixed automorphic forms of

biweightν, μwith respect to the equivariant pairρ, τand the latticeΓ.

We assert the following.

Proposition 3.2. The functional spaceMν,μ

Γ,τis nontrivial if and only if the real-valued function

1/πφν,μρ in2.6takes integral values onΓ×Γ.

Proof. The proof can be handled in a similar way as in 6 making use of 2.5combined

with the equivariant condition2.2. Indeed, assume thatMν,μ_Γ,τis nontrivial, and letFbe

a nonzero function belonging toMν,μ_Γ,τ. According to2.5, we get

Fγγ·z3.2 Fγ·γ·z3.3 Jρ,τν,μ

γ·γ, zFz

2.5

e2iφν,μρ γ,γ_Jν,μ ρ,τ

γ, γ·zJρ,τν,μ

γ, zFz,

3.4

for everyγ, γ∈Γ. On the other hand, we can write

Fγγ·zFzγγFγ·zγFγ·γ·z

3.3

Jρ,τν,μ

γ, γ·zFγ·z

3.3

Jρ,τν,μ

γ, γ·zJρ,τν,μ

γ, zFz.

3.5

Now, by equating the right hand sides of 3.4 and 3.5, keeping in mind that F is not identically zero, we get necessarily

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Conversely, by classical analysis, we pick an arbitrary nonzero C∞ and compactly supported functionψ with support contained in a fundamental domainΛΓof the lattice

Γ. Next, we consider the associated Poincar´e seriesPν,μ_Γ ψgiven by

Pν,μ

Γ ψ

z

γ∈Γ

Jρ,τν,μ

γ, zψγ·z. _3.7

Then, it can be shown that the functionP_Γν,μψisC∞and a nonzero function on forΓbeing

discrete and Suppψ ⊂ΛΓ. Indeed, for everyz∈Suppψ, we have

Pν,μ

Γ ψ

z ψz. 3.8

Furthermore, under the condition that1/πφν,μρ takes integral values onΓ×Γ, we see that

Pν,μ

Γ ψbelongs toM ν,μ

Γ,τ. In fact, for everyγ∈Γandz∈

n_{, we have}

Pν,μ

Γ ψ

γ·z

k∈Γ

Jρ,τν,μ

k, γ·zψk·γ·z

k∈Γ

Jρ,τν,μ

k, γ·zψkγ·z

hγk

h∈Γ

Jρ,τν,μ

h·γ−1, γ·zψh·z

2.5

h∈Γ

e−2iφν,μρ h,γ−1_Jν,μ ρ,τ

h, γ−1·γ·zJρ,τν,μ

γ−1, γ·zψh·z.

3.9

Finally, sincee2iφν,μρ γ,h₁_{by hypothesis on}_φν,μ

ρ , it follows that

Pν,μ

Γ ψ

γ·z

h∈Γ

Jρ,τν,μh, zJρ,τν,μ

γ−1, γ·zψh·z

Jρ,τν,μ

γ−1, γ·z

h∈Γ

Jρ,τν,μh, zψh·z

Jρ,τν,μ

γ, zPν,μ_Γ ψz.

3.10

The last equality, that is,

Jρ,τν,μ

γ−1, γ·zJρ,τν,μ

γ, z 3.11

holds for everyγ∈Γusing the chain rule2.5and taking into account the assumption made onφν,μρ . This completes the proof.

Remark 3.3. The condition involved inProposition 3.2ensures thatMν,μ_Γ,τcan be realized as

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4. On the Function

ϕ

ν,μτ

In order to prove the main result of this paper, we need to introduce the functionϕν,μτ .

Proposition 4.1. The first-order diﬀerential equation

∂ϕν,μτ

∂z −iμ

τ∂τ ∂z −τ

∂τ ∂z − ∂τ_∂z 2

−∂τ

∂z 2 z 4.1

admits a solutionϕν,μτ : → such that

ϕν,μτ is constant.

Proof. By writing theG-endomorphismρ : G → G T asρg χg, ψg, and

diﬀerentiating the equivariant condition

τg·zρg·τz χgτz ψg, 4.2

it follows that

∂g·z ∂z

∂τ ∂z

g·zχg∂τ ∂zz,

∂g·z ∂z

∂τ ∂z

g·zχg∂τ

∂zz. 4.3

Hence, for∂g·z/∂zandχgbeing inT, we deduce that

Bτν,μ

g·zνμ

∂τ_∂zg·z

2

−∂τ

∂z

g·z

2

νμ

∂τ_∂zz

2

−∂τ

∂zz

2

Bτν,μz.

4.4

Therefore,z →Bν,μτ zis a real-valued constant functionsince the onlyG-invariant functions

on are the constants. Now, by considering the diﬀerential 1-form

θτν,μz:i

νzdz−zdz μτdτ−τdτ, 4.5

one checks thatdθν,μτ dθB ν,μ

τ _{, where}_θBν,μτ _:_iBν,μ

τ zdz−zdz. Therefore, there exists a function

ϕν,μτ : → such that

ϕν,μτ Constant and satisfying the first-order partial diﬀerential

equation

∂ϕν,μτ

∂z −i

ν−Bτν,μ

zμ

τ∂τ ∂z−τ

∂τ ∂z −iμ τ∂τ ∂z −τ

∂τ ∂z − ∂τ_∂z 2

−∂τ

∂z 2 z . 4.6

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Remark 4.2. The partial diﬀerential equation4.6satisfied byϕν,μτ can be reduced further to

the following:

∂ψτν,μ

∂z τ ∂τ ∂z,

∂ψτν,μ

∂z

1

μ ν−B

ν,μ τ

zτ∂τ

∂z, 4.7

withϕν,μτ z iBν,μτ −ν|z|2−μ|τz|2 2iμψτν,μz.

5. Main Result

Letϕν,μτ be the real part ofϕ ν,μ

τ −ϕ

ν,μ

τ 0, whereϕ ν,μ

τ is a complex-valued function on as in

Proposition 4.1. DefineWν,μτ to be the special transformation given by

Wν,μ

τ

fz:eiϕτν,μz_f_z_. _5.1

We have the following.

Theorem 5.1. The image of Mν,μ

Γ,τ by the transform 5.1 is the space of LandauΓ, χτ

-auto-morphic functions. More exactly, one has

Wν,μ

τ Mν,μ_Γ,τ

F;C∞, Fzγχτ

γjBν,μτ _{γ, z}_F_z

, 5.2

withBν,μτ νμ|∂τ/∂z|2− |∂τ/∂z|2∈ andχτis the pseudocharacter defined onΓby

χτ

γexp 2iϕν,μτ

γ−2iμ

τ0, ργ−1·0. 5.3

For the proof, we begin with the following.

Lemma 5.2. The functionχτ defined on ×Γby

χτ

z;γ:eiϕν,μτ zγ−ϕν,μτ z_e−2iBν,μτ −νz,γμτz,ργ

−1_·0

5.4

is independent of the variablez.

Proof. Diﬀerentiation ofχτz;γwith respect to the variablezgives

∂χτ

∂z i

∂ϕν,μτ

∂z

zγ−∂ϕ

ν,μ τ

∂z z

χτ

−Bτν,μ−ν

γμργ−1·0∂τ

∂zz− ρ γ

−1_·₀∂τ

∂zz

χτ.

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On the other hand, using the equivariant conditionτzγ ργ·τzand4.6, one gets

i

∂ϕν,μτ

∂z

zγ− ∂ϕ

ν,μ τ

∂z z

Bτν,μγSν,μτ z−Sν,μτ

zγ

Bν,μτ −ν

γμ

aγ∂τ

∂zz−aγ ∂τ ∂zz

,

5.6

where we have setaγ ργ−1·0. Thus, from5.5and5.6, we conclude that∂χτ/∂z 0.

Similarly, one gets also∂χτ/∂z0. This ends the proof ofLemma 5.2.

Proof ofTheorem 5.1. We have to prove thatWν,μτ Fbelongs to

FBν,μτ

Γ,χτ :

F:

C∞

−→;F

zγχτ

γjBν,μτ _{γ, z}_F_z

, 5.7

wheneverF ∈ Mν,μ_Γ,τ, where

χτ

γ:exp 2iϕν,μτ

γ−2iμ

τ0, ργ−1·0. 5.8

Indeed, we have

Wν,μ

τ F

zγ:eiϕν,μτ zγ_F_z_γ

eiϕν,μτ zγ_jν_{γ, z}_jμ_ρ_γ_{, τ}_z_F_z

eiϕν,μτ zγ−ϕν,μτ z_jν_{γ, z}_jμ_ρ_γ_{, τ}_zWν,μ

τ F

z

χτ

z;γj−Bτν,μ_{γ, z}Wν,μ

τ F

z.

5.9

Whence byLemma 5.2, we see thatχτz;γ χτ0;γ :χτγ, and therefore

Wν,μ

τ F

zγχτ

γj−Bν,μτ _{γ, z}Wν,μ

τ F

z. 5.10

The proof is complete.

Corollary 5.3. The functionχτγ exp2iϕν,μτ γ−2iμτ0, ργ

−1_·₀_{satisfies the following}

pseudocharacter property:

χτ

γγe2iBν,μτ γ,γ

χτ

γχτ

γ 5.11

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Acknowledgment

The author is indebted to Professor A. Intissar for valuable discussions and encouragement.

References

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Annalen, vol. 271, no. 1, pp. 53–80, 1985.

2 P. Stiller, “Special values of Dirichlet series, monodromy, and the periods of automorphic forms,”

Mem-oirs of the American Mathematical Society, vol. 49, no. 299, p. iv116, 1984.

3 Y. Choie, “Construction of mixed automorphic forms,” The Journal of the Australian Mathematical Society.

Series A, vol. 63, no. 3, pp. 390–395, 1997.

4 M. H. Lee, “Eisenstein series and Poincar´e series for mixed automorphic forms,” Collectanea

Mathemat-ica, vol. 51, no. 3, pp. 225–236, 2000.

5 M. H. Lee, Mixed Automorphic Forms, Torus Bundles, and Jacobi Forms, vol. 1845 of Lecture Notes in

Math-ematics, Springer, Berlin, Germany, 2004.

6 A. Ghanmi and A. Intissar, “Landau automorphic forms of n_{of magnitude}_ν_{,” Journal of Mathematical}

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